Dominik Szynal

On complete convergence for arrays of rowwise independent random elements in Banach spaces

We extend and generalize some recent results on complete convergence (cf. Hu, Moricz, and Taylor [14], Gut [11], Wang, Bhaskara Rao, and Yang [26], Kuczmaszewska and Szynal [17], and Sung [23]) for arrays of rowwise independent Banach space valued random elements. In the main result, no assumptions are made concerning the existence of expected values or absolute moments of the random elements and no assumptions are made concerning the geometry of the underlying Banach space. Some well-known results from the literature are obtained easily as corollaries. The corresponding convergence rates are also established

RECURRENCE RELATIONS FOR SINGLE AND PRODUCT MOMENTS OF GENERALIZED ORDER STATISTICS FROM PARETO, GENERALIZED PARETO, AND BURR DISTRIBUTIONS

We give recurrence relations for single and product moments of generalized order statistics under the concept of Kamps from Pareto, generalized Pareto and Burr distributions. The results include as particular cases the above relations for moments of k–th record values.

On complete convergence in a Banach space

Sufficient conditions are given under which a sequence of independent random elements taking values in a Banach space satisfy the Hsu and Robbins law of large numbers. The complete convergence of random indexed sums of random elements is also considered.

Recurrence relations for single and product moments of k-th record values from pareto, generalized pareto and burr distributions

In this note we give recurrence relations satisfied by single and product momenrs of k-th upper-record values from the Pareto, generalized Pareto and Burr distributions. From these relations one can obtain all the single and product moments of all k-th record values and at the same time all record values ( k=1). Moreover, we see that the single and product moment of all k-th record values from these distributions can be exprrssed in terms of the moments of the minimal statistic of a k-sample from the exponential distribution may be deduced by letting the shape parameter deptend to 0. At the end we give characterizations of the three distributions considered. These results generalize, among other things, those given by Balakrishnan and Abuamllah (1994).

On characterization of certain distributions of kth lower (upper) record values

General classes of continuous distributions are characterized by the conditional expectation of the kth lower record values. Specific distributions considered as particular cases of the general class of distributions are inverse exponential, inverse Weibull, inverse Pareto, negative exponential, negative Weibull, negative Pareto, negative power, Gumbel, exponentiated-Weibull, loglogistic, Burr X, inverse Burr XII and inverse paralogistic distributions.

Moments of certain Deformed probability distributions

Abstract We consider properties of two classes of discrete probability distributions, namely the so-called Deformed Modified Factorial Series Distributions (DMFSD) and Deformed Modified Power Series Distributions (DMPSD). The formulae for moments and recurrence relations for the moments of these deformed distributions are derived. The results obtained generalize or extend some Theorems given by Janardan (Janardan, K. G. (1984). Moments of certain series distributions and their applications. SIAM J. Appl. Math. 44:854–868) and Gupta (Gupta, R. C. (1974). Modified power series distributions and some of its applications. Sankhyã, Ser. B 35:288–298).

On existence,asymptotic behaviour and stability of solutions of stochastic integral equations

This paper deals with the equation where H is an adapted corlol process, Z is a semimartingale such that Zo=0 a.s.and satisfies some regularity conditions. There are given sufficient conditions for the existence of random solutions and their asymptotic behaviour. We base on the notion of a measure of noncompactness in Banach space and the fixed–point theorem of Darbo type Results of this paper generalize those of [5],[7]

On existence and asymptotic behaviour of solutions of a nonlinear stochastic integral equation

SummaryThe aim of the paper is to study a nonlinear stochastic integral equation of the form

On a characterization of optimal predictors for nonstationary ARMA processes

x(t;\omega ) = h(t,x(t;\omega )) + \int\limits_0^t {k_1 (t,\tau ;\omega )f_1 \left( {\tau ,x(\tau ;\omega )} \right)d\tau } + \int\limits_0^t {k_2 (t,\tau ;\omega )f_2 \left( {\tau ,x(\tau ;\omega )} \right)d\beta (\tau ;\omega )} .